Model Checking for CTL

1
Model Checking for CTL

• Marks the states of K by subformulas of P
• s is marked by a subformula Q if Q holds
• at TK,s
• The algorithm proceeds from simple formulas
• to more complex formulas for all states
  simultaneously.

2
Algorithm

• For atomic formulas immediately
• For Boolean connectives easy
• s is marked by P1 P2 if .
• For modal connectives P1 ?U P2 if from s there
  is a P1 path to a P2 node.
• For modal connectives P1 ?U P2

3
CTL

• Modalities E( a formula of TL(U))
• A ( a formula of TL(U))
• Semantics T,s E C if there is a path from s
• which has a property C.

4
Model Checking for CTL

• How to check E ( property of a path)
• Construct an automaton A for the property.
• Take the product with the Kripke Structure.

5
Equation for P1 ?U P2

• X - the set that satisfy P1 ?U P2
• X ?? P2 ? ?? (X P1 )
• XH(X) where H ? Y. ?? P2 ? ?? (Y P1 )
• How many solution ZH(Z) has?

6
Characterization of P1 ?U P2

• P1 ?U P2 is the minimal solution of
• Z ?? P2 ? ?? (Z P1 )
• X0 ?? P2
• Xn1 ?? P2 ? ?? (Xn P1 )
• s in Xn iff there is a P1 path of length n1
  from s to P2
• X ? Xn XH(X) and H monotonic

7
Mu-calculus

• E At At X E1 E2 E1?E2
• ?? E A? E µ X. E ?X.E
• Semantics µ least fixed point ? greatest fixed
  point.
• E ? the set of states that satisfies E in
  the enviroment ? Var-gt States.

8
EGp

• EGp ?X.p ?? X

9
From mu-calculus to MLO

• Theorem for every mu-formula c(X1,,Xn)
• there is an MLO formula b(t, X1,Xn) which
• is equivalent to c over trees.
• Theorem for every future MLO formula b(t,X1,Xn)
  which is invariant under counting there is an
  equivalent (over trees) mu formula c.

10
Symbolic Model Checking

• Explicit Model Checking
• Input a finite state K and a formula c
• Task Find the states of K that satisfy c.
• Symbolic model checking
• Input a description of K and a formula c
• Task Find a description of the states of K
  that satisfy c.

11
A description of Kripke structures by formulas

• s(x1,,xn) describes a set of states
• t(x1,xn,x1,xn) describes transitions
• For every label p a formula lp(x1,xn) that
  describes the states labeled by p.

12
BDT, and OBDD

• Binary decision trees
• Ordered Binary Decision Diagrams.